6. Constitutive Laws

 

The equations listed in the preceding sections apply to any solid or fluid that deforms under the action of external forces.   They do not, by themselves, have a unique solution, however, because the deformations are not related in any way to internal forces.

 

The governing equations are completed by constitutive laws that provide the missing connection.  Unlike deformation measures; kinetics; and conservation laws, a constitutive law cannot be calculated or predicted from first principles, except for a few very special cases such as small deformations of crystalline materials, where elastic properties can be estimated using ab-initio techniques that approximate quantum mechanical level atomic scale interactions in some way.

 

6.1 Basic assumptions [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa [email protected]@  local homogeneity of internal forces and deformation; the principle of local action

 

Before discussing specific constitutive models, it is helpful to review the basic assumptions that we take for granted in developing stress-strain laws.  They are listed below.

 A very small sample that is extracted from the solid has uniform properties;

 When the solid is deformed, initially straight lines in the solid are deformed into smooth curves (with continuous slope);

 This means that very short line segments (much shorter than the radius of curvature of the curves) are just stretched and rotated by the deformation.   Consequently, the deformation of a sufficiently small volume element can be characterized by the deformation gradient;

 The stress at a point in the solid depends only on the change in shape of a vanishingly small volume element surrounding the point.  It must therefore be a function of the deformation gradient or a strain measure that is derived from it.

If we accept the preceding assumptions, it means that we can measure the relationship between stress and strain by doing an experiment that induces a uniform strain in a suitable sample of the material.  According to our assumptions, the stress should also be uniform, and can be calculated from the forces acting on the specimen.

 

These are clearly approximations.  Materials are not really uniform at small scales, whether you choose to look at the atomic scale, or the microstructural scale.  However, these features are usually much smaller than the size of the solid part or component, and the material can be regarded as statistically uniform, in the sense that if you cut two specimens with similar size out of the material they will behave in the same way.  A continuum model then describes the average stress and deformation in a region of the material that is larger than microstructural features, but small compared with the dimensions of the part.

 

 

 

6.2 Restrictions on constitutive equations

 

You may be called upon to develop a stress-strain law for a new material at some point of your career.  If so, it is essential to make sure that the stress-strain law satisfies two conditions:

(i) It must obey the laws of thermodynamics.

(ii) It must satisfy the condition of objectivity, or material frame indifference

In addition, it is a good idea to ensure that the material satisfies the Drucker stability criterion discussed in more detail below.  Of course, your proposed law must conform to experimental measurements, and if possible should be based on some understanding of the physical processes that govern the response of the solid.   But only the first two conditions apply to all solids.  

 

Thermodynamic restrictions: Constitutive laws usually start by expressing the specific internal or free energy, specific entropy, and heat flux of a material in terms of the temperature, parameters characterizing shape changes, and any internal state variables (such as yield stress) that characterize the material state.  These have the general form

 

Internal energy: ε(θ,deformation measures,state variables) Entropy: s(θ,deformation measures,state variables) Free energy: ψ(θ,deformation measures,state variables) = εθs Heat flux response function: q(θ,deformation measures,state variables) Stress Response Function:  σ(θ,deformation measures,state variables) Evolution equations for state variables (these may be functions of temperature, kinematics, and state vars) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaeysaiaab6gacaqG0bGaaeyzai aabkhacaqGUbGaaeyyaiaabYgacaqGGaGaaeyzaiaab6gacaqGLbGa aeOCaiaabEgacaqG5bGaaeOoaiaabccacqaH1oqzcaGGOaGaeqiUde NaaiilaiaabsgacaqGLbGaaeOzaiaab+gacaqGYbGaaeyBaiaabgga caqG0bGaaeyAaiaab+gacaqGUbGaaeiiaiaab2gacaqGLbGaaeyyai aabohacaqG1bGaaeOCaiaabwgacaqGZbGaaeilaiaabohacaqG0bGa aeyyaiaabshacaqGLbGaaeiiaiaabAhacaqGHbGaaeOCaiaabMgaca qGHbGaaeOyaiaabYgacaqGLbGaae4CaiaabMcaaeaacaqGfbGaaeOB aiaabshacaqGYbGaae4BaiaabchacaqG5bGaaeOoaiaabccacaWGZb GaaiikaiabeI7aXjaacYcacaqGKbGaaeyzaiaabAgacaqGVbGaaeOC aiaab2gacaqGHbGaaeiDaiaabMgacaqGVbGaaeOBaiaabccacaqGTb GaaeyzaiaabggacaqGZbGaaeyDaiaabkhacaqGLbGaae4CaiaabYca caqGZbGaaeiDaiaabggacaqG0bGaaeyzaiaabccacaqG2bGaaeyyai aabkhacaqGPbGaaeyyaiaabkgacaqGSbGaaeyzaiaabohacaqGPaaa baGaaeOraiaabkhacaqGLbGaaeyzaiaabccacaqGLbGaaeOBaiaabw gacaqGYbGaae4zaiaabMhacaqG6aGaaeiiaiabeI8a5jaacIcacqaH 4oqCcaGGSaGaaeizaiaabwgacaqGMbGaae4BaiaabkhacaqGTbGaae yyaiaabshacaqGPbGaae4Baiaab6gacaqGGaGaaeyBaiaabwgacaqG HbGaae4CaiaabwhacaqGYbGaaeyzaiaabohacaqGSaGaae4Caiaabs hacaqGHbGaaeiDaiaabwgacaqGGaGaaeODaiaabggacaqGYbGaaeyA aiaabggacaqGIbGaaeiBaiaabwgacaqGZbGaaeykaiaabccacaqG9a Gaaeiiaiabew7aLjabgkHiTiabeI7aXjaadohaaeaacaqGibGaaeyz aiaabggacaqG0bGaaeiiaiaabAgacaqGSbGaaeyDaiaabIhacaqGGa GaaeOCaiaabwgacaqGZbGaaeiCaiaab+gacaqGUbGaae4Caiaabwga caqGGaGaaeOzaiaabwhacaqGUbGaae4yaiaabshacaqGPbGaae4Bai aab6gacaqG6aGaaeiiaiaahghacaGGOaGaeqiUdeNaaiilaiaabsga caqGLbGaaeOzaiaab+gacaqGYbGaaeyBaiaabggacaqG0bGaaeyAai aab+gacaqGUbGaaeiiaiaab2gacaqGLbGaaeyyaiaabohacaqG1bGa aeOCaiaabwgacaqGZbGaaeilaiaabohacaqG0bGaaeyyaiaabshaca qGLbGaaeiiaiaabAhacaqGHbGaaeOCaiaabMgacaqGHbGaaeOyaiaa bYgacaqGLbGaae4CaiaabMcaaeaacaqGtbGaaeiDaiaabkhacaqGLb Gaae4CaiaabohacaqGGaGaaeOuaiaabwgacaqGZbGaaeiCaiaab+ga caqGUbGaae4CaiaabwgacaqGGaGaaeOraiaabwhacaqGUbGaae4yai aabshacaqGPbGaae4Baiaab6gacaqG6aGaaeiiaiaabccacaWHdpGa aiikaiabeI7aXjaacYcacaqGKbGaaeyzaiaabAgacaqGVbGaaeOCai aab2gacaqGHbGaaeiDaiaabMgacaqGVbGaaeOBaiaabccacaqGTbGa aeyzaiaabggacaqGZbGaaeyDaiaabkhacaqGLbGaae4CaiaabYcaca qGZbGaaeiDaiaabggacaqG0bGaaeyzaiaabccacaqG2bGaaeyyaiaa bkhacaqGPbGaaeyyaiaabkgacaqGSbGaaeyzaiaabohacaqGPaaaba GaaeyraiaabAhacaqGVbGaaeiBaiaabwhacaqG0bGaaeyAaiaab+ga caqGUbGaaeiiaiaabwgacaqGXbGaaeyDaiaabggacaqG0bGaaeyAai aab+gacaqGUbGaae4CaiaabccacaqGMbGaae4BaiaabkhacaqGGaGa ae4CaiaabshacaqGHbGaaeiDaiaabwgacaqGGaGaaeODaiaabggaca qGYbGaaeyAaiaabggacaqGIbGaaeiBaiaabwgacaqGZbGaaeiiaiaa bIcacaqG0bGaaeiAaiaabwgacaqGZbGaaeyzaiaabccacaqGTbGaae yyaiaabMhacaqGGaGaaeOyaiaabwgacaqGGaGaaeOzaiaabwhacaqG UbGaae4yaiaabshacaqGPbGaae4Baiaab6gacaqGZbGaaeiiaiaab+ gacaqGMbGaaeiiaiaabshacaqGLbGaaeyBaiaabchacaqGLbGaaeOC aiaabggacaqG0bGaaeyDaiaabkhacaqGLbGaaeilaiaabccacaqGRb GaaeyAaiaab6gacaqGLbGaaeyBaiaabggacaqG0bGaaeyAaiaaboga caqGZbGaaeilaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaae4Cai aabshacaqGHbGaaeiDaiaabwgacaqGGaGaaeODaiaabggacaqGYbGa [email protected]@

 

For a solid with mass density (in the deformed configuration) ρ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  experiencing a stretch rate D ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae [email protected]@  and Cauchy stress σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa [email protected]@ , the first and second laws of thermodynamics then require that

ρ ε t | x=const = σ ij D ij q i y i +q [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaeiaabaWaaSaaaeaacqGHci ITcqaH1oqzaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaa hIhacqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccq GH9aqpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamiramaa BaaaleaacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaaiabgkGi2k aadghadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSba [email protected]@

σ ij D ij 1 θ q i θ y i ρ( ψ t +s θ t )0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTmaa laaabaGaaGymaaqaaiabeI7aXbaacaWGXbWaaSbaaSqaaiaadMgaae qaaOWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgkHiTiabeg8aYnaabmaabaWaaSaaae aacqGHciITcqaHipqEaeaacqGHciITcaWG0baaaiabgUcaRiaadoha daWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi2kaadshaaaaacaGLOa [email protected]@

for all possible processes.

 

In practice, only the second law leads to any restrictions on the constitutive equations.  We assume that the first law can always be satisfied by some appropriate heat flux q.

 

We will see how to use the second law in devising structures for constitutive equations through specific examples in the next few sections.  In practice, a simplified version of the second law is used nearly always used, however, and we summarize this form briefly here.

 

For the majority of problems in solid and fluid mechanics, we do not attempt to model deformation and heat conduction simultaneously (there are exceptions of course, such as high strain rate deformation; thermoelasticity, and so on).   In practice, we can often assume that the solid is in equilibrium with a surrounding heat bath with constant temperature, and heat flow through a solid is sufficiently rapid to ensure that the temperature remains approximately uniform.  Under these conditions the second law reduces to

σ ij D ij ρ ψ t 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiab eg8aYnaalaaabaGaeyOaIyRaeqiYdKhabaGaeyOaIyRaamiDaaaacq [email protected]@

Physically, this states that the rate of work done by stresses must always equal or exceed the rate of change of free energy of the solid. 

 

Frame indifference:  The principle of frame indifference (which is still the subject of some debate) states that constitutive equations must be invariant to a change of observer (all observers, regardless of their reference frame, must describe the same physical process).   The transformation of various field quantities under changes of observer was discussed in the preceding section.   To reiterate, spatial coordinates map as

y * = y 0 * (t)+Q(t)(y y 0 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWH5bWaa0baaSqaaiaaicdaaeaacaGGQaaaaOGaaiikaiaa dshacaGGPaGaey4kaSIaaCyuaiaacIcacaWG0bGaaiykaiaacIcaca [email protected]@

under a change of observer.   In a constitutive model, observers in the two frames will  use some temperature or deformation measures that we will denote by a vague symbol Γ, Γ * [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo5ahjaacYcacqqHtoWrdaahaaWcbe [email protected]@  (they could be scalar, vector, or tensor valued variables).  These measures are always related by a appropriate transformation relations under a change of observer [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  for example, deformation gradients, Lagrange strains, and the Left stretch tensor transform according to F ij * = Q ik F kj E ij * = E ij B ij * = Q ik B kl Q jl [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaqhaaWcbaGaamyAaiaadQgaae aacaGGQaaaaOGaeyypa0JaamyuamaaBaaaleaacaWGPbGaam4Aaaqa baGccaWGgbWaaSbaaSqaaiaadUgacaWGQbaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caWGfbWaa0baaSqaaiaadMgacaWGQbaabaGaaiOk aaaakiabg2da9iaadweadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOqamaaDaaaleaa caWGPbGaamOAaaqaaiaacQcaaaGccqGH9aqpcaWGrbWaaSbaaSqaai aadMgacaWGRbaabeaakiaadkeadaWgaaWcbaGaam4AaiaadYgaaeqa [email protected]@ .   The constitutive model will predict some physical quantity of interest in terms of these variables.  We will denote the predicted quantity in the two frames by a similar vague variable Φ, Φ * [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agjaacYcacqqHMoGrdaahaaWcbe [email protected]@ , which could be a scalar, vector or tensor valued quantity.   For example, the constitutive model could predict the free energy ψ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , the heat flux vector q i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  or Cauchy or Material stress σ ij , Σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa [email protected]@ .   Again, we know how these physical quantities transform under a change of observer [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@ ψ= ψ * q i * = Q ik q i σ ij * = Q ik σ kl Q jl Σ ij * = Σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iabeI8a5naaCaaale qabaGaaiOkaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caWGXbWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaeyypa0 JaamyuamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGXbWaaSbaaSqa aiaadMgaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqaHdpWCdaqhaaWcbaGaamyAaiaadQgaaeaacaGG QaaaaOGaeyypa0JaamyuamaaBaaaleaacaWGPbGaam4AaaqabaGccq aHdpWCdaWgaaWcbaGaam4AaiaadYgaaeqaaOGaamyuamaaBaaaleaa caWGQbGaamiBaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abfo6atnaaDaaaleaacaWGPbGaamOAaaqaaiaacQcaaaGccqGH9aqp [email protected]@ .    The functions relating these  variables to the kinematic variables must behave correctly under a change of observer [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  in general

Φ=f(Γ) Φ * =f( Γ * ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agjabg2da9iaadAgacaGGOaGaeu 4KdCKaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab gsDiBlaaykW7caaMc8UaaGPaVlaaykW7cqqHMoGrdaahaaWcbeqaai aacQcaaaGccqGH9aqpcaWGMbGaaiikaiabfo5ahnaaCaaaleqabaGa [email protected]@

As a specific example, a constitutive relation specifying the Cauchy stress in terms of deformation gradient would have to transform according to

σ ij = f ij ( F ij ) σ ij * = f ij ( F ij * ) Q ik σ kl Q jl = f ij ( Q ik F kj ) Q ik f kl ( F ij ) Q jl = f ij ( Q ik F kj ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aa0baaSqaaiaadMgaca WGQbaabaaaaOGaeyypa0JaamOzamaaBaaaleaacaWGPbGaamOAaaqa baGccaGGOaGaamOramaaDaaaleaacaWGPbGaamOAaaqaaaaakiaacM cacaaMc8Uaeyi1HSTaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGa aiOkaaaakiabg2da9iaadAgadaWgaaWcbaGaamyAaiaadQgaaeqaaO GaaiikaiaadAeadaqhaaWcbaGaamyAaiaadQgaaeaacaGGQaaaaOGa aiykaiaaykW7aeaacqGHuhY2caWGrbWaaSbaaSqaaiaadMgacaWGRb aabeaakiabeo8aZnaaBaaaleaacaWGRbGaamiBaaqabaGccaWGrbWa aSbaaSqaaiaadQgacaWGSbaabeaakiabg2da9iaadAgadaWgaaWcba GaamyAaiaadQgaaeqaaOGaaiikaiaadgfadaWgaaWcbaGaamyAaiaa dUgaaeqaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqaaaaakiaacM cacqGHuhY2caWGrbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadAga daWgaaWcbaGaam4AaiaadYgaaeqaaOGaaiikaiaadAeadaWgaaWcba GaamyAaiaadQgaaeqaaOGaaiykaiaadgfadaWgaaWcbaGaamOAaiaa dYgaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGPbGaamOAaaqaba GccaGGOaGaamyuamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGgbWa [email protected]@

This restricts the possible form of the function f ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaamyAaiaadQgaae [email protected]@ .   You can use a similar process to crank out restrictions for any kind of constitutive relation.  

 

There is still some debate as to whether frame indifference should apply under a transformation between frames in relative rotation.   Usually, it is assumed that constitutive equations must hold even in a non-inertial frame.   But some fluid mechanics models of turbulent flow, for example, attempt to describe macroscopic behavior of a volume element of fluid that has internal angular momentum.   Constitutive laws of this kind can only hold in an inertial frame, and so invariance need only apply during a transformation from one inertial frame to another.   These have the form

y * = y 0 * +Q(y y 0 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWH5bWaa0baaSqaaiaaicdaaeaacaGGQaaaaOGaey4kaSIa aCyuaiaacIcacaWH5bGaeyOeI0IaaCyEamaaBaaaleaacaaIWaaabe [email protected]@

(the time dependence is gone).   

 

 

Drucker Stability:  For most practical applications, the constitutive equation must satisfy a condition known as the Drucker stability criterion, which can be expressed as follows.  Consider a deformable solid, subjected to boundary tractions t i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , which induce some displacement field u i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ .  Suppose that the tractions are increased to t i +Δ t i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaamyAaaqabaGccq [email protected]@ , resulting in an additional displacement Δ u i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadwhadaWgaaWcbaGaamyAaa [email protected]@ . The material is said to be stable in the sense of Drucker if the work done by the tractions Δ t i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadshadaWgaaWcbaGaamyAaa [email protected]@  through the displacements Δ u i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadwhadaWgaaWcbaGaamyAaa [email protected]@  is positive or zero for all Δ t i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadshadaWgaaWcbaGaamyAaa [email protected]@ :

ΔW= { A Δ t i dΔ u i dt } dt0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadEfacqGH9aqpdaWdbaqaam aacmaabaWaa8quaeaacqqHuoarcaWG0bWaaSbaaSqaaiaadMgaaeqa aOWaaSaaaeaacaWGKbGaeuiLdqKaamyDamaaBaaaleaacaWGPbaabe aaaOqaaiaadsgacaWG0baaaaWcbaGaamyqaaqab0Gaey4kIipaaOGa ay5Eaiaaw2haaaWcbeqab0Gaey4kIipakiaadsgacaWG0bGaeyyzIm [email protected]@

You can show that this condition is satisfied as long as the stress-strain relation obeys

Δ τ ij Δ ε ij 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabes8a0naaBaaaleaacaWGPb GaamOAaaqabaGccqqHuoarcqaH1oqzdaWgaaWcbaGaamyAaiaadQga [email protected]@

where Δ τ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabes8a0naaBaaaleaacaWGPb [email protected]@  is the change in Kirchhoff stress, and Δ ε ij =(Δ u i / x j +Δ u j / x i ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcaGGOaGaeyOaIyRaeuiLdqKaamyDamaa BaaaleaacaWGPbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaai aadQgaaeqaaOGaey4kaSIaeyOaIyRaeuiLdqKaamyDamaaBaaaleaa caWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadMgaae [email protected]@  is an increment in strain resulting from an infinitesimal change in displacement Δ u i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadwhadaWgaaWcbaGaamyAaa [email protected]@ .

 

This is not a thermodynamic law (the work done by the change in tractions is not a physically meaningful quantity), and there is nothing to say that real materials have to satisfy Drucker stability. In fact, many materials show clear signs that they are not stable in the sense of Drucker.   However, if you try to solve a boundary value problem for a material that violates the Drucker stability criterion, you are likely to run into trouble.  The problem will probably not have a unique solution, and in addition you are likely to find that smooth curves on the undeformed solid develop kinks (and may not even be continuous) after the solid is deformed.  This kind of deformation violates one of the fundamental assumptions underlying continuum constitutive equations. 

A simple example of a stress-strain curve for material that is not stable in the sense of Drucker is shown in the picture.  The stability criterion is violated wherever the stress decreases with strain in tension, or increases with strain in compression. For the former we see that Δ σ 11 <0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeo8aZnaaBaaaleaacaaIXa [email protected]@ , while Δ ε 11 >0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLnaaBaaaleaacaaIXa [email protected]@ ; for the latter Δ σ 11 >0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeo8aZnaaBaaaleaacaaIXa [email protected]@ , while Δ ε 11 <0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLnaaBaaaleaacaaIXa [email protected]@ .